# How to calculate 3 sigma

Written by john lister
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Sigma is another term for standard deviation, which is a measure of how much variance there is in a set of data. For example, in a class of five-year-old children the standard deviation in height would likely be low; in a national Olympics team with basketball players and weightlifters, the standard deviation in height would likely be high. Standard deviation is calculated by a set mathematical formula and can then be used as a reference point and a way of checking data.

Skill level:
Moderately Easy

## Instructions

1. 1

Calculate the mean average of all the data points in your set. For example, add up every child's height in inches and divide by the number of children.

2. 2

Calculate the variance for each data point (the difference between the data point and the average), then square the result and write it in a separate list. For example, if the first child's height is 48 inches and the average height of the class is 50 inches, the variance is minus two. The square of minus two is minus four, so add this to the new list. Repeat for each data point -- in this case, each child's height.

3. 3

Calculate the mean average of the new list. Take care as some of the figures will be positive and some negative. The result given by the mean average is called the variance.

4. 4

Calculate the square root of the mean average. The result is the standard deviation, or one sigma. Multiply by three to get the three sigma result for the group.

#### Tips and warnings

• Statistically, on average, 68 percent of results should fall into one standard deviation or sigma. For example, if the average salary of a group you survey is £50,000 and the standard deviation is £5,000, then 68 percent of people in the group should have a salary between £45,000 and £55,000. In a similar way, 95 percent of results should fall into two sigma and 99.7 percent should fall into three sigma. In this example, that means 99.7 percent of people in the group should have a salary between £35,000 and £65,000, meaning up to three times £5,000 (£15,000) above of below the average of £50,000.
• If your actual results vary significantly from this pattern it is a sign, but not a guarantee, that something is wrong with your data, particularly if it comes from a sample group designed to represent a larger population. It could be that the group you used is too small, the group does not accurately represent the larger population (for example, you calculated the average height of a sports centre user by carrying out a survey just after a basketball game), or that you made a mistake in collecting or calculating the data.

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